Question

Let $I={\int }_1^3\sqrt{3+x^3}dx$, then the value of I lies in the interval

• A. [4, 6]
• B. [1, 3]
• C. $[4,2\sqrt{30}]$
• D. $[\sqrt{15},\sqrt{30}]$

Answer 1

Answer: (A), (C)

The correct options are
A

[4, 6]

C

$[4,2\sqrt{30}]$

$f\left(x\right)=\sqrt{(3+x^3)}$ this is increasing function on [1, 3]

least value (m) = f(1) = $\sqrt{(3+1^3)}=2$ and

greatest value $M=f\left(3\right)=\sqrt{(3+3^3)}=\sqrt{30}$

So $(3-1)2\le {\int }_1^3\sqrt{(3+x^3)}dx\le (3-1)\sqrt{30}$

$4\le {\int }_1^3\sqrt{3+x^3}dx\le 2\sqrt{30}$